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A waiting time phenomenon for thin film equations. (English) Zbl 1024.35051
The authors consider the fourth-order degenerate parabolic equation $u_t + \nabla \cdot (|u|^n \nabla \Delta u)=0, \quad t>0, \;x \in \Omega, \tag{1}$ where $$\Omega \subseteq \mathbb{R}^N$$, $$N \in \{ 1,2,3\}$$. Equation (1) is supplemented with suitable initial condition $$u_0$$ with compact support strictly contained in $$\Omega$$ and Neumann boundary conditions on $$u$$ and $$\Delta u$$ on $$\partial \Omega$$. Specifically, for $$n \in (0,3)$$ they ask the following question: Can one formulate conditions on the initial data $$u_0$$ so that the solution of (1), $$u(t)$$ exhibits the waiting time phenomenon, which in the case of $$N=1$$ means the following: Let $$x_0 \in \partial \operatorname {supp} u_0$$. Then $$u(t)$$ exhibits the waiting time phenomenon at $$x_0$$ if there is a time $$T^*$$ and a neighbourhood $$B(x_0)$$ of $$x_0$$ such that $$\operatorname {supp} u(t) \cap B(x_0) = \operatorname {supp} u_0 \cap B(x_0)$$; for the multidimensional version (employing an external cone condition) see section 5 of the paper.
The main results of the paper are: For $$0<n<2$$ and $$N=1$$ or if $$1/8<n<2$$, $$N=2,3$$, the waiting time phenomenon occurs at a point $$x_0 \in \partial\operatorname {supp} u_0$$ if $$u_0(x)$$ grows at most as $$|x-x_0|^{4/n}$$ in a neighbourhood of $$x_0$$ (Theorems 4.1 and 5.1). For $$2\leq n < 3$$, $$N=1$$ the waiting time phenomenon occurs at $$x_0$$ if $$u_0x$$ grows ar most like $$|x-x_0|^{4/n-1}$$ (Theorem 6.1). Finally, in section 7 optimality of these exponents is considered and results of numerics suggesting such optimality are presented.
The main tools used in the proofs are a version of the Gagliardo-Nirenberg inequalities (Theorem 2.3), entropy (for $$n<2$$) and weighted energy estimates (for $$n \geq 2$$), and a crucial extension of an iteration lemma due to Stampacchia (Lemma 3.1).

##### MSC:
 35K65 Degenerate parabolic equations 35K35 Initial-boundary value problems for higher-order parabolic equations 35R35 Free boundary problems for PDEs 76A20 Thin fluid films 74K35 Thin films
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